QM : Transition Prob -> Decay Rate

1. The problem statement, all variables and given/known data
Its not a specific homework problem, buta general problem that a friend and I keep arguing about it.

Assume you have a system of:
N Particles.
2 Possible States, A && B
@ t=0 all N particles are in state A (such that c_a*c_a = P_a = 1)
where c_a is the coefficient for state A of a single particle.

Now the coefficient for state B is time varying. Technically it sort of goes like (1/f[t]) Sin^2 [wt], but that should be unnecessary.

If a particle is measured to be in state "B" then lets say that it is taken out of the group and the process is restarted, or it keeps going or whatever. But that particle is taken out. So N-1 particles remain.

My question is, how can I define a rate at which N is changing as a function of time.
dN/dt? How fast is the system degrading?

It seems logically I would have to come up with some way of knowing how often a measurement is taken in order to know how it evolves. (Collapsing the wavefunctions)

If I "set the removal tool on auto, with continuous monitoring", walk away for 10 minutes, is it possible to know how much smaller the group has become? Or is it only possible to know that "At some time "t" there is a probability associated with each possible value of "N" left"

Or is the rate just defined as the change in the probability as a function of time?
Like R = dP/dt = (d(c_b)^2/dt) ??